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Crystal Structures
Types of crystal structures
Face centered cubic (FCC)
Body centered cubic (BCC) Hexagonal close packed (HCP)
Close Packed Structures
Dierent Packing of HCP and FCC
Crystallograpic Directions and Planes
cubic syste!s
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Face Centered Cubic (FCC)
"to!s are arranged at te cornersand center of eac cube face of te
cell# "to!s are assu!ed to touc along face
diagonals
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Face Centered Cubic (FCC)
Te lattice para!eter$ a$ is related to teradius of te ato! in te cell troug%
Coordination nu!ber% te nu!ber ofnearest neigbors to any ato!# For FCCsyste!s$ te coordination nu!ber is &'#
22 Ra =
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Face Centered Cubic (FCC)
"to!ic Packing Factor% te ratio ofato!ic spere olu!e to unit cell
olu!e$ assu!ing a ard spere!odel#
FCC syste!s ae an "PF of #*+$ te
!axi!u! packing for a syste! in ,icall speres ae e-ual dia!eter#
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Body Centered Cubic
"to!s are arranged at te corners ofte cube ,it anoter ato! at te
cube center#
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Body Centered Cubic
Since ato!s are assu!ed to toucalong te cube diagonal in BCC$ te
lattice para!eter is related to ato!icradius troug%
3
4 R
a =
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Body Centered Cubic
Coordination nu!ber for BCC is .#/ac center ato! is surrounded by
te eigt corner ato!s# Te lo,er coordination nu!ber also
results in a sligtly lo,er "PF for BCC
structures# BCC as an "PF of #0.$rater tan #*+ in FCC
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Hexagonal Close Packed
Cell of an HCP lattice is isuali1ed asa top and botto! plane of * ato!s$
for!ing a regular exagon around acentral ato!# 2n bet,een teseplanes is a alf3exagon of 4 ato!s#
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Hexagonal Close Packed
Tere are t,o lattice para!eters in HCP$ a
and c$ representing te basal and eigtpara!eters respectiely# 2n te ideal case$
te c/a ratio is ,$ o,eer$ deiations dooccur#
Coordination nu!ber and "PF for HCP areexactly te sa!e as tose for FCC% &' and#*+ respectiely#
Tis is because tey are bot considered closepacked structures#
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Close Packed Structures
/en toug FCC and HCP are closepacked structures$ tey are -uite dierentin te !anner of stacking teir closepacked planes#
Close packed stacking in HCP takes placealong te c direction ( te (&) plane)# FCCclose packed planes are along te (&&&)#
First plane is isuali1ed as an ato! surroundedby 0 nearest neigbors in bot HCP and FCC#
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Close Packed Structures
Te second plane in bot HCP and FCC issituated in te 5oles6 aboe te 7rst plane ofato!s#
T,o possible place!ents for te tird plane ofato!s
Tird plane is placed directly aboe te 7rst planeof ato!s
8 "B" stacking 33 HCP structure Tird plane is placed aboe te 5oles6 of te 7rst
plane not coered by te second plane8 "BC stacking 33 FCC structure
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Close Packed Structures
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Crystallographic
Directions
Cubic syste!s
directions are na!ed based upon te
pro9ection of a ector fro! te origin of tecrystal to anoter point in te cell#
Conentionally$ a rigt and Cartesiancoordinate syste! is used#
Te cosen origin is arbitrary$ but isal,ays selected for te easiest solution tote proble!#
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Crystallographic
Directions
Points ,itin te lattice are ,ritten inte for! $k$l$ ,ere te tree
indices correspond to te fraction ofte lattice para!eters in te x$y$1direction#
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Miller Indices
Procedure for ,riting directions in:iller 2ndices
Deter!ine te coordinates of te t,opoints in te direction# (Si!pli7ed if oneof te points is te origin)#
Subtract te coordinates of te second
point fro! tose of te 7rst# Clear fractions to gie lo,est integer
alues for all coordinates
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Miller Indices
2ndices are ,ritten in s-uare brackets,itout co!!as (ex% ;kl ten te direction is
][ kl h
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Miller Indices
Crystallograpic Planes
2dentify te coordinate intercepts of te
plane te coordinates at ,ic te plane intercepts
te x$ y and 1 axes#
2f a plane is parallel to an axis$ its intercept istaken as ∞#
2f a plane passes troug te origin$ coose ane-uialent plane$ or !oe te origin
Take te reciprocal of te intercepts
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Miller Indices
Clear fractions due to te reciprocal$ but do notreduce to lo,est integer alues#
Planes are ,ritten in parenteses$ ,it bars
oer te negatie indices# /x% (kl) or if > ten it beco!es
ex% plane " is parallel to x$ and intercepts yand 1 at &$ and terefore is te (&&)#
Plane B passes troug te origin$ so teorigin is !oed to ?@$ tereby !aking teplane te
)( kl h
)121(
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Miller Indices